Finite Group Databasebeta

← #27170440

Subgroups — #27170440

Order 12928.

228 conjugacy classes of subgroups; only the 45 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 45 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20265919C211yesmaximal, characteristic
2#20265571C211yesmaximal, characteristic, complemented
3#20265537C211yesmaximal, characteristic, complemented
4#20265536C211yesmaximal, characteristic, complemented
5#20264916C211yesmaximal, characteristic, complemented
6#20264916C211yesmaximal, characteristic, complemented
7#329×C101C211yesmaximal, characteristic, complemented
8C2×Dic404—21no—
9C2×#13108063—21no—
10#15692255C2211yescharacteristic, complemented
11#15692244—21no—
12#15692243C2211yescharacteristic
13#15692243C2211yescharacteristic
14#15692241—21no—
15#15692241—21no—
16#15692225—21no—
17#15692225—21no—
18#15692223—21no—
19#15692180—21no—
20#15692179—21no—
21#15692177—21no—
22#15692176—21no—
23#15692166C2211yescharacteristic, complemented
24#124×C101C2211yescharacteristic
25#97×C101C2211yescharacteristic, complemented
26#97×C101C2211yescharacteristic, complemented
27C23×D101—21no—
28C2×Dic202—21no—
29C2×Dic202D411yescharacteristic
30C2×Dic202D411yescharacteristic
31C2×D404—21no—
32C2×D404D411yescharacteristic
33C2×D404D411yescharacteristic
34C2×C22⋊D101—21no—
35C2×C22⋊D101—21no—
36Dic404—42no—
37#13108063—42no—
38C2×#11717579—21no—
39C4⋊Dic101—21no—
40C4⋊Dic101—21no—
41C22⋊Dic101—21no—
42#13108051—21no—
43#13108051—21no—
44#13108051—21no—
45#13108051—21no—
46#13108051—21no—
47#13108051—21no—
48C4×Dic101—21no—
49C2×Q8×C101D411yescharacteristic
50C2×D4×C101D411yescharacteristic
51C2×C8×C101—21no—
52C2×C8×C101—21no—
53C22⋊C4×C101—21no—
54C22⋊C4×C101—21no—
55C42×C101C2311yescharacteristic
56C23×C101—21no—
57C22×D101—21no—
58C22×D101—21no—
59C22×D101—81no—
60C22×D101—81no—
61C22×D101—81no—
62Q8×C101—41no—
63D4×C101—41no—
64C2×C4×C101—21no—
65C2×C4×C101C2×D411yescharacteristic
66C2×C4×C101C2×D411yescharacteristic
67C2×C4×C101C2×D411yescharacteristic
68C22⋊D101—81no—
69C22⋊D101—81no—
70C2×Dic101—21no—
71C2×Dic101—21no—
72C2×Dic101—41no—
73D404—41no—
74D404—41no—
75D404—81no—
76Dic202—41no—
77Dic202—41no—
78Dic202—41no—
79Dic202—41no—
80C8×C101—22no—
81C8×C101—41no—
82#11717579—41no—
83C22×C101—41no—
84C22×C101—41no—
85C22×C101—41no—
86C22×C101#12011yescharacteristic
87C2×D101—41no—
88C2×D101—41no—
89C2×D101—41no—
90C2×D101—41no—
91C2×D101—41no—
92C2×D101—41no—
93C2×D101—84no—
94C4×C101—21no—
95C4×C101—21no—
96C4×C101—21no—
97C4×C101—41no—
98Dic101—41no—
99Dic101—41no—
100Dic101—81no—
101C2×C101—81no—
102C2×C101#45311yescharacteristic
103C2×C101#45511yescharacteristic
104C2×C101#45511yescharacteristic
105D101—81no—
106D101—81no—
107#1659—1011nomaximal, Sylow
108C101#165911yescharacteristic, complemented, Sylow
109H(3,𝔽4)—1011no—
110#497—1011no—
111#489—1011no—
112#489—1011no—
113#329—1011no—
114#329—1011no—
115#329D10111yescharacteristic, complemented
116C2×Q16—2021no—
117C2×SD16—2021no—
118C4⋊Q8—1011no—
119#124—1011no—
120#124—1011no—
121#124—2021no—
122#124—2021no—
123#124—2021no—
124#124C2×D10111yescharacteristic
125#120—2021no—
126#120—2021no—
127#120—2021no—
128#103—2021no—
129#103—2021no—
130#102—2021no—
131#102—2021no—
132#97—1011no—
133#97C2×D10111yescharacteristic, complemented
134#97C2×D10111yescharacteristic, complemented
135C24—2021no—
136C2×Q8—1011no—
137C2×Q8—1011no—
138C2×Q8—2021no—
139C2×Q8D40411yescharacteristic
140C2×D4—1011no—
141C2×D4—1011no—
142C2×D4—2021no—
143C2×D4—2021no—
144C2×D4—2021no—
145C2×D4D40411yescharacteristic
146Q16—4042no—
147SD16—4042no—
148C2×C8—21no—
149C2×C8—21no—
150C2×C8—2021no—
151C4⋊C4—2021no—
152C4⋊C4—2021no—
153C22⋊C4—21no—
154C22⋊C4—21no—
155C22⋊C4—2021no—
156C22⋊C4—2021no—
157C22⋊C4—2021no—
158C22⋊C4—2021no—
159C22⋊C4—2021no—
160C22⋊C4—2021no—
161C22⋊C4—2021no—
162C42—2021no—
163C42C22×D10111yescharacteristic
164C23—21no—
165C23—8081no—
166C23—8081no—
167C23—8081no—
168C23—2021no—
169C23—2021no—
170Q8—41no—
171Q8—4041no—
172Q8—4041no—
173Q8—4041no—
174Q8—4041no—
175D4—41no—
176D4—8081no—
177D4—8081no—
178D4—8081no—
179D4—4041no—
180D4—4041no—
181C2×C4—21no—
182C2×C4—4041no—
183C2×C4—2021no—
184C2×C4—2021no—
185C2×C4D4×D10111yescharacteristic
186C2×C4D4×D10111yescharacteristic
187C2×C4C2×D40411yescharacteristic
188C8—22no—
189C8—41no—
190C8—4041no—
191C22—41no—
192C22—41no—
193C22—41no—
194C22—8084no—
195C22—4041no—
196C22—4041no—
197C22—4041no—
198C22—4041no—
199C22—4041no—
200C22—4041no—
201C22#1569222511yescharacteristic
202C4—21no—
203C4—21no—
204C4—21no—
205C4—41no—
206C4—8081no—
207C4—4041no—
208C4—4041no—
209C2—81no—
210C2—8081no—
211C2—8081no—
212C2#2026534611yescharacteristic
213C2#2026535611yescharacteristic
214C2#2026535711yescharacteristic
subgroupG/H
centerC22#15692225
commutator, cosocleC42×C101C23
FrattiniC42C22×D101
Fitting#329×C101C2
socleC22×C101#120
prime radical#20265919C2
hypercenter#329D101
hypocenterC101#1659
LCSG ▷ C42×C101 ▷ C22×C101 ▷ C101
UCSC1 ◁ C22 ◁ C42 ◁ #329

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. #20265571C2semidirect
3. #20265537C2semidirect
4. #20265536C2semidirect
5. #20264916C2semidirect
6. #20264916C2semidirect
7. #329×C101C2semidirect
10. #15692255C22semidirect
23. #15692166C22semidirect
25. #97×C101C22semidirect
26. #97×C101C22semidirect
108. C101#1659semidirect
115. #329D101semidirect
133. #97C2×D101semidirect
134. #97C2×D101semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#202659192
#202655712
#202655372
#202655362
#202649162
#329×C1012
#1659101